# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### A variation of Cesaro means

It follows from the Cesaro means that $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n g(k)=c~~~{\rm if}~~~\lim_{k\rightarrow\infty}g(k)=c.$$ Instead, let us consider a sequence $\{f_n\}$ such that ...
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### $E_n - A$ invertible if $\|A \|< 1$

Let $E_n$ be the identity matrix and $\|\cdot \|$ a matrix norm. How to prove with the help of Banach's fixed-point theorem that $E_n - A$ is invertible if $\|\,A\,\| < 1$?
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### Existence of a function that fulfills an equation

I am just revising for my exams and came across this question: Show that in the Banach-space of functions that are continuous in the interval $[-1,1]$, together with the supremum-norm, there is ...
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### Uniform continuity supremum

In the User's Guide to Viscosity Solutions, it is claimed at the top of page 31 that If $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ is a uniformly continuous function, there exists a positive ...
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### Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
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### Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
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### Math needed to study Navier-Stokes existence and smoothness problem [closed]

This is Navier-Stokes existence and smoothness problem. I think the main problem is that I am not familiar with the mathematics of the Navier-Stokes existence smoothness problem. What kind of math ...
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### Suppose a sequence's subsequences have at least one subsubsequence that converges almost surely to $X$. Prove convergence in probability

Probability with Martingales What I tried: 'only if' Suppose a sequence converges in probability to $X$. By $d$ there exists a subsequence that converges almost surelyto $X$. Then by $a$, ...
Let $f:X\rightarrow X$ a homeomorphism where $X$ is a compact metric space. Fix $x\in X$, denote $O(f,x)=\{ f^n(x):n\in \mathbb{Z}\}$ the orbit of $f$ by $x$. For $m\in \mathbb{N}$ denote $O(f,x,m)... 0answers 28 views ### Estimate the Sobolev norm of negative order of a function I would like to estimate the Sobolev norm of order$-1$of the function$f(x)$which is defined as follows: Let$\psi\in C^\infty_c(\mathbb{R})$be a compactly supported smooth function on the real ... 1answer 21 views ### This is a question about the best type of regression analysis to use in my software Let me start by saying that I am not a mathematician and I am not very good at math. I am mainly interested in obtaining the best possible results. I am currently doing trial and error with my ... 1answer 113 views ### What branch/field of mathematics is this? [closed] I do not want solutions, I just want the field/branch of mathematics that these problems deal with, and possibly a good online source or two to learn it. Problems :- 1:- 2:- 3:- 4:- ... 1answer 39 views ### A continuous onto/surjective function from$[0, 1) \to \Bbb R$. Does there exist a continuous onto/surjective function from$[0, 1) \to \Bbb R$? Finding difficult to site an example... 2answers 72 views ### What is the diagonal principle? I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of$(c)$). What is the diagonal principle? Is that related to Cantor's ... 4answers 54 views ### How to show that$x \sin \frac{\pi}{x} > \pi \cos \frac{\pi}{x}$for$x \in (1, \infty)$? How could I prove that$x \sin \frac{\pi}{x} > \pi \cos \frac{\pi}{x}$for$x \in (1, \infty)$? Dividing both sides through by$x \sin \frac{\pi}{x}$and letting$y = \frac{\pi}{x}$gives the ... 1answer 28 views ### Maximum variance Consider a random variable$X$with continuous probability density$f(x)$and compact support, say$[a,b]$with$a<b$. Moreover, let$f(x)$vanish at the boundary, i.e.$f(a) = f(b) = 0$. ... 1answer 35 views ###$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$. Show that$F(\cdot , t_0)$has exactly one zero using the Implicit Function Theorem$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$is a through$t$parametrized family of polynominals.$a_i : I \to \Bbb R \:\:\:\mathrm{ are }\: \mathcal C^k$- functions with$k \ge 1$. Let$x_0$be a zero ... 2answers 42 views ###$n\lvert a_n\rvert \to 0$, then$\sum_{k=1}^n \frac k n\lvert a_k\rvert \to 0$? I am now working on the converse of Abel's THM and found out one proof of the conditional converse of the theorem. The proof says, Suppose $$n\lvert a_n\rvert \to 0$$ as $$n \to \infty$$ Then, $$\... 0answers 34 views ### Computing \delta as a function of \epsilon for uniformly continuous functions So a function is uniformly continuous if \forall \epsilon > 0, \exists \delta >0 s.t. \forall x,y, |x-y|<\delta \rightarrow |f(x)-f(y)| < \epsilon. Is there another category of ... 1answer 10 views ### Finding a 2-chain whose boundary is c - c_{1, n} for a 1-cube c such that c(0) = c(1) and a circle c_{1, n}. Problem (Spivak, 4-24). Define c_{1, n} : [0, 1] \to \mathbb{R}^2 by c_{1, n}(t) = (\cos 2\pi nt, \sin 2\pi nt). If c is a singular 1-cube in \mathbb{R}^2-\{ 0 \} with c(0) = c(1), show ... 2answers 57 views ### Is this set countable or uncountable? (Related to mean value theorem) Let f be a differentiable function on the real line. Consider any h\in (0,1). By the mean value theorem there exists d_h\in (0,h) such that f'(d_h)=\frac{f(h)-f(0)}{h}. Is the set \{d_h\} ... 1answer 70 views ### Construction of a measure space from some weird functional Here is the complete problem I am trying to solve, but currently, I am just interested in proving that \Sigma is a \sigma-algebra. Let X be a set and \phi: 2^X \to [0, \infty] be a ... 1answer 31 views ### Smooth Approximations in L^2((0,1)) Let L^2((0,1)) be as usual the Lebesgue space of measurable complex-valued functions f:(0,1) \rightarrow \mathbb{C} such that \int |f(x)|^2 dx < \infty. It is a well known fact (see e.g Lieb ... 0answers 22 views ### A C^1 function in Orlicz Sobolev space How to prove that thi functional is C^1:$$ I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx $$Where \Phi is an N-function and F(t)=\int_{0}^t f(s) ds where ... 1answer 79 views ### Good, relatively short math textbooks? [closed] Recently I've been trying to decide on some fun math summer reading on some areas of math which I have less experience with. I'm an undergrad studying mathematics with a focus in actuarial science, ... 1answer 119 views ### Closure of \frac {1} {n} [duplicate] I have as a definition of the closure of a set E in a metric space (S,d) that E^- is the intersection of all closed sets containing E (Elementary analysis the theory of calculus by Kenneth Ross).... 1answer 53 views ### Why is sum \cot^{-1}(n^2+n+1) equal to \cot^{-1}\left(\frac1{m+1}\right)? Can u Please Explain This sum result :$$\sum_{n=0}^m \cot^{-1} (n^2+n+1) = \cot^{-1} \left(\frac1{m+1}\right)$$1answer 33 views ### f \colon \mathbb{R}^n \to \mathbb{R}^n of class C^1 such that ||f(x) - f(y)|| \geq c||x-y|| is a diffeomorphism. Let f \colon \mathbb{R}^n \to \mathbb{R}^n be a C^1 map. Suppose that exists c>0 such that ||f(x)-f(y)||\geq c||x-y|| for all x,y \in \mathbb{R}^n. Prove that f is a diffeomorphism. I ... 1answer 35 views ### How prove that the range of an rectifiable curve has measure zero? Let f:[a,b] \rightarrow \mathbb{R}^n be a rectifiable continuous curve, show that f[a,b] has content zero. If f is continuous then is integrable so we have that [a,b]\times f[a,b] has ... 1answer 25 views ### Integral on \partial B(0,R) I want to calculate \int_{\partial B(x,\epsilon)}\frac{1}{|x-y|^2} with B(x,\epsilon)\in \mathbb{R}^3 Time ago I saw a paper who said (if I am right) \int_{\partial B(0,R)}\frac{1}{|x-y|^{n-1}}=... 3answers 64 views ### Finding nth derivative in an unusual way If f(z) = \frac{e^{iz}}{z^2-1}, then f^{(4)}(z) can be found by differentiating f(z) four times. I tried to use Cauchy's integral formula, but the integrand is not holomorphic at z=0, so we ... 2answers 29 views ### f(x)=x\ln x-\frac{k}{x} and f(x_1)=f(x_2)=0 \Rightarrow f'\left(\frac{x_1+x_2}{2}\right)\not=0 If f(x)=x\ln x-\frac{k}{x}. And x_1, x_2 are two roots of f(x)=0. Then f'\left(\frac{x_1+x_2}{2}\right)\not=0 First, I determine the range of k. Because f=0 has two roots \iff x^2\... 1answer 31 views ### Differentiable, nonnegative derivative ae, nondecreasing? I have been trying to prove or disprove: Let g be differentiable on the reals, have g'(x)\geq 0 except countably many values, then g is non-decreasing. The main problem I face is that I don't ... 1answer 25 views ### Finding Taylor series without using derivatives If \displaystyle f(z) = \frac{e^{iz}}{z^2-1} then we can set g(z)=e^{iz} and h(z)=z^2-1. The Maclaurin expansion for e^{iz} is$$\sum\limits_{n=0}^\infty \frac{(iz)^n}{n!}$$so \displaystyle ... 1answer 44 views ### Is there a function f satisfying this integrability condition? I am wondering if it is possible to find a real valued function f such that$$ \int_{0}^{1} \frac{1}{x^2 T(f(x))} \ dx < \infty $$where T(x)=\cos(x) or T(x)=\sin(x). Thanks. 2answers 49 views ### How did Euler prove the partial fraction expansion of the cotangent function? As far as we know, Euler was the first to prove$$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$I've seen several modern proofs of it and they all ... 2answers 90 views ### Nonexistence of a continuous injection$f:S^2 \rightarrow \mathbb{R^2}$What is the "easiest" way to show that there is no continuous injection$f:S^2 \rightarrow \mathbb{R^2}$? Sure the Borsuk-Ulam theorem implies that result, but this may be a "difficult" way. 0answers 25 views ### Product of p-summable sequences Let$1<p<\infty$and$(a_n)_{n=0}^\infty$and$(b_n)_{n=0}^\infty$be two complex,$p$-summable sequences, which means that \sum_{n=1}^\infty |a_n|^p < \infty, \sum_{n=1}^\... 1answer 28 views ### Theorem on arithmetic of natural numbers. From "Analysis I"-Herbert & Joachim: (starting from the Peano axioms) "There are operations addition + , multiplication · and a partial order ≤ on N which are uniquely determined by the ... 2answers 35 views ### To show$f$conatined in oval is constant Let$\Omega \subset \mathbb{C}$is connected open and let$f\in O(\Omega)$Suppose$f(\Omega) \subset L$where$L:=\{x+iy \in \mathbb{C} \vert x^{2k} + y^{2k} = 1 \}$for$k \in \mathbb{Z}^{+}, k >...
this This is analysis exam problem. If $\sum_n{x_n}$ converges, then $\sum_n{x_n\over n}$ converges. I don't know how to show that.